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Consider two curves 81 2 12 and 82 21112 forming a coordinate system: Draw the grid foned by 81 and 82 for iso-contours of of 1,2,3,4,and 5. Find unit vectors in th...

Question

Consider two curves 81 2 12 and 82 21112 forming a coordinate system: Draw the grid foned by 81 and 82 for iso-contours of of 1,2,3,4,and 5. Find unit vectors in this coordinate system. Show this coordinate system in orthogonal. Find the gradient operator in this coordinate system

Consider two curves 81 2 12 and 82 21112 forming a coordinate system: Draw the grid foned by 81 and 82 for iso-contours of of 1,2,3,4,and 5. Find unit vectors in this coordinate system. Show this coordinate system in orthogonal. Find the gradient operator in this coordinate system



Answers

Orthogonal plane Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes $2 x+5 y-3 z=0$ and $-x+5 y+2 z=8$

Yeah. So in this question, we have been given the values of Vector you and Victor V, and we need to prove whether tha two actors are or O'Connell. Let us first find out the dark product off you and we shall be equal toe eight I minus four days and to minus six. I minus 12 j. So this will be a quarter to eight into minus six plus minus four in two, minus 12. So this will be a call to zero. Now, we know that the angle T tablet in the two actress has given us cause and worse off. You'd RV divided by magnitude off you into magnitude of we now, since you don t zero so this whole town will evaluate to zero so we can write three ties the quarto cause in worse off zero, which is the quarto 90 degrees. Since the ties equal to 90 degrees. Therefore directors are orthogonal. Yeah,

We have our number 60 60. We have to. We use graphical utility to escape the intersecting graphs, all the questions and to show that there are terminals. So our plan should be The graphs will be orthogonal if the tangents at the points of intersections will be perpendicular. So we need to find slopes off both the attendants. Tenders on both the cubs are the points off intersections. So let us write equations first. Why, Squire? Equal to x Q and do X squared plus three y square equal toe five. No. Yeah. I have used graphical utility toe draw the graph, Which is the Christmas. Okay, Yes. This is the graph off these equations. One is, uh, elliptical equation that the lips. Okay, this is a question. This is why I exist. This is X success. And this is a graph of this ellipse, and another is graph of this equation. Now it is quite clear that points of intersection, sir. One comma, one and minus one. I'm sorry. One call minus one. Okay. One comma, minus one. So we have to find the slopes of attendance at both these points. Okay. So let us, uh, find you already expressed D over DX for Why Squire equal to x. Q. Let us differentiate both sides with respect to X de wever d X x Q. This will be to y d Y by D. X equal to three X square. So let us divide both sides by two. I will be getting Do you ever the X equal toe three X squared by the wind. So let us assume this to be M one that is, slope off the slow for the attendant expression for snowboard attendant for this equation. Now for second question, we have to differentiate both sides with spec tracks again. The body X on two X Square plus three y square equal toe DVD X five. So this will be four x plus six y d y by D. X equal to zero. Let us subtract both sides a little subject for extra board decides when we're getting six way The Vibe R D X equal to minus four x not divide both sides by six wives will be getting. Do you have any eggs equal to minus four X My six way This will be to works. This will be three by so the whereby. The Excel Vehicle toe minus two x by three way. The decision is to be M two that is, expression off the slopes off tangent at and for after the Ellipse two X squared, plus serious, quite well defined. Now let us start with the point when first point is one comma one so m one that is value off the first it contends. Slope of the tangent three access squired by two y three x Esquire by two y that is three into one Squire by 21 That is three by two AM to value off a stroke attendant at this point minus two x by y so minus two into one by three and 21 So this reminders to three we have just plug in the value of X and y here. Okay, so now if we calculate m one into em too, that is three by two and two minus two by three. This is minus one since slope multiplication on that, the product off slopes are the same. Ar minus one product off the slopes is minus one, so slopes are perpendicular, so curb is orthogonal at one comma. One now comes by a second point. That is, of one comma minus one. In this case. Yeah. Okay, So when that is, slope off Tencent for who was quite equal to X Q at 0.1 comma minus one three x Inspired by two y. That is three X Esquire by tour. At one comma minus one. That is three and two. Yeah, by two into one. That is to into minus 13 by minus two. That is minus three by two. I am, too. Is the global tangent at one comma, minus one for the ellipse that will be minus two Expert three wives, minus two weeks. My three way that is minus two into one by three and two minus one. That is two or three. Now if you take product off a morning to em too. So that will be minus three by two. Into two by three, two and two will get canceled out. Three and several it cancel out. We look getting minus one, which means he has seen slope multiplication off the slopes of the attendants is equal to minus one. So slopes are or weakens. Attend himself perpendicular. So the cops are or so Gunnell at one comma, minus one and at all the points of intersections. Thank you.

Hello there. So for days exercise we got these two vectors U. And V. We're going to define the inner product with respect to the matrix A. So it is basically multiplying each of the vectors with the matrix A. And then take the usual the product that we know. So in this case the Matrix A. is the finest 2111. And we need to show that you is a phenomenal to be you remember the two vectors are functional in some space within their products. If and only if the inner product of those two vectors is equal to zero. So this is what we need to show. So let's start by computing the multiplication of the matrix by the factors in this case is 2111 time is the vector 33 And this is just the vector 9, 6. And likewise for we we have 21 11 Times a vector five -8. And this is just the venture two minus three. So this is what we need. And then we just need to calculate the inner product of these two factors. That's going to be just the product of 9, 6 With 2 -3. And the result of this is 18 minus 18, Which is clearly zero. Therefore you and we are, you know.

So for this problem we are given a hyperbole to with the equation X squared minus Y squared is equal to fight. And and it lives with the equation for X squared plus nine. Y squared is equal to 72. And what we wanted to do is to prove that These two curves are diagonal and we say that to cribs are talking about it at each point of intersection. The angle between them is 90° or right angle. So first let's try to identify the expression for the slope of tangents. At for each of these two curves hyper hyperbole. A. And lips for the hyperbole to we have to XD explains. Do Y. Dy let me try to write that is equal to zero and dy over DX is equal to is simply equal to two x over two y. Or simply X over Y. Yeah. Now for let's call it um some point this is X over Y. Before they left we'll have eight x. d. x plus 18 Y. Dy zero. That being said the way over the actually simply equal to negative negative B. Picks Over nine Y. or simply -4 X. Sorry, should be eating 84 x over name one. The next thing I wanted to find out or the intersection The intersection of these two groups. Mhm. And how are we going to do that? So from the first equation we say that X squared is equal to fight plus Y squared, plugging in the planning, plugging this equation to the ellipse equation. We have four X squared which is equipment in the five. That's why scream that's nine Y squared is equal to 72 will end up having 20 That's four y squared. This name was squared is equal to 72 or 13. Y squared is equal to 52, dividing both sides by 13 will have voice were disappointed four or why is simply equal to plus remind us two, solve for X. Using this equation, X squared is equal to five plus widespread. Why squared is equal to four coming from this one? So we say X squared is simply equal to nine or X is able to plus or minus three. Hey, no, there are multiple points right here. So the point of intersection czar three comma two, three comma negative too -3 camera too. In 83 camera -2. And for each this of this point will try to um analyze whether the slope of these two curbs are actually perpendicular by the way. Let me just say here AM two is equal to negative four X over the main worry. So let's start with the first point To a three camera too. M one Is equal to three hubs. Mhm M two is equal to -2/3 in take note stick note of this perpendicular, perpendicular lines are said to have the following slope relationship. M1 is able to the negative simple hall of the slope of the other line and as you can see here, that relationship is actually observant between the scopes of these two curves at that point. So we can say that yes they are perpendicular at this point of intersection. Now let's try a three. come on 82. So we'll have someone is able to negative rehabs M two becomes two thirds. And this religion right here still um it can still be seen on this M one and M. Two a cat specific point of intersection. Now let's try to do it for the next two point of intersection. Negative three comma too. So we'll have em one is equal to negative three hubs. I am two ish still two thirds. And then finally for -3 come and eat it too. We have em one equals 3 hubs & M two is equal to negative two thirds. And for all of this point of intersection, the relationship of slope M one and M two are actually um observe but to obey this relationship right here and that being said because he that these curves are actually or talking with one another


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